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📅 May 5, 2026 👤 scraface

May 05, 2026

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📌 2\sin x + 3\cos x = R\sin(x + \phi)

📌 where $ R = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} $.

📌 2\sin x + 3\cos x = \sqrt{13} \sin(x + \phi)

📌 The range of $ \sqrt{13} \sin(x + \phi) $ is $ [-\sqrt{13}, \sqrt{13}] $.

📌 2\sin x + 3\cos x + 4 \in [4 - \sqrt{13}, 4 + \sqrt{13}]

📌 Since $ \sqrt{13} \approx 3.605 $, $ 4 - \sqrt{13} \approx 0.395 > 0 $, so the expression never reaches zero.

📌 Therefore, the absolute value $ |2\sin x + 3\cos x + 4| $ achieves its minimum at the minimum of the continuous function on an interval where it is always positive.

📌 The minimum occurs when $ 2\sin x + 3\cos x $ is minimized, but since it's always greater than $ -\sqrt{13} $, and greater than $ -4 $, the expression inside is always positive.

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